Question

which equation is represented by the graph below? the graph is a coordinate plane with x-axis from -5 to 5 and y-axis from -5 to 5. the curve passes through (0,1) and increases rapidly. the options are: y = ln x, y = ln x + 1, y = e^x, y = e^x + 1

Explanation:

Step1: Analyze the domain of the graph

The graph is defined for all real numbers (x can be negative, zero, positive), while logarithmic functions (\(y = \ln x\), \(y=\ln x + 1\)) have a domain of \(x>0\), so we can eliminate the first two options.

Step2: Check the y - intercept

The y - intercept occurs when \(x = 0\). For \(y=e^{x}\), when \(x = 0\), \(y=e^{0}=1\). For \(y = e^{x}+1\), when \(x = 0\), \(y=e^{0}+1=2\). From the graph, when \(x = 0\), \(y = 1\), so the function is \(y = e^{x}\).

Answer:

\(y = e^{x}\) (the option with \(y = e^{x}\))